Sorry, Seaman Rogers, I'll fix it.

If Blythe is alone, Ensign Rivers refers to her, if Claire is alone, Ensign Rivers refers to her.  If they're together, people will say Ensign Claire Rivers or Ensign Blythe Rivers.  This is only when speaking formally, officially, issuing orders, etc.  When people are speaking casually or in the heat of battle, they will probably use first names.